Question

Is the function f(x)= 9/x continuous for all x ?

A. f(x) is not continuous for all x because f(x) is not defined at x=9 and lim f(x) does not exist.
x->9

B. f(x) is not continuous for all x because f(x) is not defined at x=0 and lim f(x) does not exist.
x->0

C. f(x) is continuous for all x because f(x) is defined for all x and lim f(x) = 0.
x->0

D. f(x) is continuous for all x because f(x) is defined for all x and lim f(x) =9.
x->0

E. f(x) is continuous for all x because f(x) is defined at x=9 and lim f(x) =1.
x->9

**please explain :) thank you!!

Answer 1

Well think about it, can you have a 0 in the denominator?

Answer 2

no, it would be undefined right?

Answer 3

and where it says x --> number, that's supposed to be under the lim, but it got misaligned haha sorry about that!

Answer 4

Correct. And the limit there would not exist:

\[\LARGE \lim_{x \rightarrow 0^-}~f(x)=-\infty \]
and
\[\LARGE \lim_{x\rightarrow 0^+}~f(x)=\infty\]